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The sum of the terms of a sequence is called a series.
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A geometric series is the sum of a geometric sequence.
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A geometric series with π terms can be written as π π is equal to π plus ππ plus ππ squared plus ππ cubed, and so on until you reach ππ π minus one, where π is the first term and π is the common ratio, and the common ratio is the number you multiply one term by to get to the next term in the sequence, but π cannot be one.
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Before we look at the question, weβre actually gonna have a look at this statement that π cannot be equal to one.
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The common ratio cannot be equal to one because if it was, youβd have a sequence that looks like this, which means that actually if you had π, π times one, π times one squared, π times one cubed, it would always be π, so every term would be π, so actually it would not be a sequence; itβd just be a repetition of a number.
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Okay, now letβs look at the last part of the question: find the sum of the first six terms of a geometric series with π is equal to 24 and π is equal to a half.
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To enable us to solve this problem, we need to use this formula, which tells us that the sum of the first π terms is equal to π multiplied by one minus π to the power of π over one minus π, again with the parameter that π cannot equal to one.
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And we discussed one reason why it wouldnβt work if youβre looking at each term of a sequence, but also if you look at the equation.
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The denominator one minus π would be equal to zero, and obviously we canβt have that because it wouldnβt give us a real solution.
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When solving a problem like this, weβre gonna use a formula.
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I like to write down the values we have and those weβre looking for.
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So first of all, we know that π is equal to 24, so our first term is 24.
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And we also have π is equal to a half, so we know that the common ratio is equal to a half.
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Weβll also need to find π, as you can see is in our formula, so π is equal to six because this is the number of terms.
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And if we look at our question, it says the first six terms.
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And finally, the sum of the first six terms, which we write as π with six, is what weβre looking to find in the question.
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Now we have our values; we can substitute them into the formula to find the sum of the first six terms, which gives us π six or the sum of the first six terms is equal to 24 multiplied by one minus a half to the power of six over one minus a half.
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Weβre gonna simplify this, which gives us 189 over eight divided by a half, which gives us 189 over four, cause remembering, quick tip, dividing by a half is the same as multiplying by two.
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Then finally simplify it by converting into a mixed number; it gives us the sum of the first six terms of the geometric series is 47 and a quarter.
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Again the reason thatβs 47 and a quarter is that four goes into 189 47 times with one leftover, so it gives us 47 and a quarter.